How to store area of each quad mesh from a mesh?
To output the areas of each face to a file, you would need to use a python script. It is easy to use python to create and write to a file. You can use str.format() to control how the data looks in the file.

The area is a property of each face in the objects mesh data, faces are called polygons here.

import bpy

with open('face_data.txt','w') as out_file:
    for f in bpy.context.object.data.polygons:
        out_file.write('{}, {}\n'.format(f.index, f.area))